Skip to main content
Springer Nature Link
Log in

Superstatistics: theory and applications

  • Original article
  • Published:
Continuum Mechanics and Thermodynamics Aims and scope Submit manuscript

Abstract.

Superstatistics is a superposition of two different statistics relevant to driven nonequilibrium systems with a stationary state and intensive parameter fluctuations. It contains Tsallis statistics as a special case. After briefly summarizing some of the theoretical aspects, we describe recent applications of this concept to three different physical problems, namely a) fully developed hydrodynamic turbulence, b) pattern formation in thermal convection states, and c) the statistics of cosmic rays.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Tsallis, C.: Possible generalization of Boltzmann-Gibbs statistics. J. Stat. Phys. 52, 479-487 (1988)

    MathSciNet  MATH  Google Scholar 

  2. Tsallis, C., Mendes, R.S., Plastino, A.R.: The role of constraints within generalized nonextensive statistics. Physica A 261, 534-554 (1998)

    Article  Google Scholar 

  3. Tsallis, C.: Nonextensive statistics: theoretical, experimental and computational evidence and connections. Braz. J. Phys. 29, 1-35 (1999)

    Google Scholar 

  4. Abe, S., Okamoto, Y. (eds): Nonextensive Statistical Mechanics and Its Applications, Springer Berlin (2001)

  5. Beck, C., Lewis, G.S., Swinney, H.L.: Measuring nonextensitivity parameters in a turbulent Couette-Taylor flow. Phys. Rev. E 63, 035303(R) (2001)

    Article  Google Scholar 

  6. Beck, C.: On the small-scale statistics of Lagrangian turbulence. Phys. Lett. A 287, 240-244 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  7. Daniels, K.E., Beck, C., Bodenschatz, E.: Defect turbulence and generalized statistical mechanics. cond-mat/0302623, Physica D (to appear) (2003)

  8. Bediaga, I., Curado, E.M.F., de Miranda, J.M. A thermodynamic equilibrium approach in \(e^+e^-\to\) hadrons. Physica A 286, 156-163 (2000)

    Google Scholar 

  9. Tsallis, C., Anjos, J.C., Borges, E.P.: Fluxes of cosmic rays: a delicately balanced stationary state. Phys. Lett. 310, 372-376 (2003)

    Article  Google Scholar 

  10. Wilk, G., Wlodarczyk, Z.: Interpretation of the nonextensivity parameter q in some applications of Tsallis statistics and Levy distributions. Phys. Rev. Lett. 84, 2770-2773 (2000)

    Article  Google Scholar 

  11. Beck, C.: Dynamical foundations of nonextensive statistical mechanics. Phys. Rev. Lett. 87, 180601 (2001)

    Article  Google Scholar 

  12. Beck, C., Cohen, E.G.D.: Superstatistics. Physica A 322, 267-275 (2003)

    Article  MATH  Google Scholar 

  13. Tsallis, C., Souza, A.M.C.: Constructing a statistical mechanics for Beck-Cohen superstatistics. Phys. Rev. E 67, 026106 (2003)

    Article  Google Scholar 

  14. Abe, S.: Generalized entropy optimized by an arbitrary distribution. cond-mat/0211437 (2002)

  15. Abe, S.: Stability of Tsallis entropy and instabilities of Renyi and normalized Tsallis entropies: a basis for q-exponential distributions. Phys. Rev. E 66, 046134 (2002)

    Article  Google Scholar 

  16. Tsallis, C., Souza, A.M.C.: Stability of the entropy for superstatistics. cond-mat/0301304 (2003)

  17. Reynolds, A.M.: On the application of nonextensive statistics to Lagrangian turbulence. Phys. Fluids 15, L1-L4 (2003)

  18. Beck, C.: Lagrangian acceleration statistics in turbulent flows. cond-mat/0212566 Europhys. Lett. (to appear) (2003)

  19. Castaing, B., Gagne, Y., Hopfinger, E.J.: Velocity probability density functions of high Reynolds-number turbulence. Physica D 46, 177-200 (1990)

    Article  MATH  Google Scholar 

  20. Beck, C. Superstatistics in hydrodynamic turbulence. physics/0303061 (2003) Physica D (to appear)

  21. Sattin, F., Salasnich, L.: Multiparameter generalization of nonextensive statistical mechanics. Phys. Rev. E 65, 035106(R) (2002)

    Article  Google Scholar 

  22. La Porta, A., Voth, G.A., Crawford, A.M., Alexander, J., Bodenschatz, E.: Fluid particle accelerations in fully developed turbulence. Nature 409, 1017 (2001)

    Article  Google Scholar 

  23. Voth, G.A., Measurement of particle accelerations in fully developed turbulence. J. Fluid Mech. 469, 121-160 (2002)

    Google Scholar 

  24. Daniels, K.E., Bodenschatz, E.: Statistics of defect motion in spatio-temporal chaos in inclined layer convection. Chaos 13, 55-63 (2003)

    Article  Google Scholar 

  25. Beck, C.: Non-extensive statistical mechanics and particle spectra in elementary interactions. Physica A 286, 164-180 (2000)

    Article  Google Scholar 

  26. Gaisser, T.K., Stanev, T.: Cosmic rays. Phys. Rev. D 66, 010001-1 (2002) (http://pdg.lbl.gov)

    Article  Google Scholar 

  27. Kaniadakis, G.: Statistical mechanics in the context of special relativity. Phys. Rev. E 66, 056125 (2002)

    Article  Google Scholar 

  28. Beck, C.: Generalized statistical mechanics of cosmic rays. cond-mat/0301354 (2003) Physica A (to appear)

  29. http://astroparticle.uchicago.edu/announce.htm (data compilation by Swordy, S.P.)

  30. Hagedorn, R.: Nuovo Cimento 3, 147 (1965)

Download references

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, Queen Mary, University of London, Mile End Road, E1 4NS, London, UK

    C. Beck

Authors
  1. C. Beck
    View author publications

    You can also search for this author inPubMed Google Scholar

Corresponding author

Correspondence to C. Beck.

Additional information

Communicated by M. Sugiyama

Received: 13 March 2003, Accepted: 27 May 2003, Published online: 9 December 2003

PACS:

05.40.-a

Rights and permissions

Reprints and permissions

About this article

Cite this article

Beck, C. Superstatistics: theory and applications. Continuum Mech. Thermodyn. 16, 293–304 (2004). https://doi.org/10.1007/s00161-003-0145-1

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00161-003-0145-1

Keywords: